CCS Discrete III Professor: Padraic Bartlett Homework 15: Spectral Graph Theory Due Friday, Week 9 UCSB 2015 Choose three problems below to solve! 1. Given a n × n matrix A, the trace of A is the sum of the entries on the diagonal of A: that is, n X tr(A) = A(i; i): i=1 (a) Suppose that A is a n × n matrix that has n eigenvalues λ1; : : : λn (counted with Pn multiplicity.) Prove that tr(A) = i=1 λi. (b) Let G be a finite graph with n × n adjacency matrix AG and eigenvalues λ1; : : : λn (where each eigenvalue is listed as many times as its multiplicity indicates.) 2 2 Prove that λ1 + : : : λn is a nonnegative integer. What very simple graph property does this sum correspond to? 2. Prove or disprove: there is no graph with −1=2 as an eigenvalue. 3. Let Kn;m be the complete bipartite graph on (n; m) vertices (i.e. Kn;m consists of two parts, one with n vertices and the other with m vertices. Every possible edge from one part to the other exists, and no edges exist within one given part.) Find the spectra of Kn;m. 4. Prove or disprove the following claim: there is some way to partition the edges of the graph K10 into three groups, each isomorphic to copies of the Petersen graph. 5.A Latin square is a n × n array filled with the symbols f1 : : : ng, so that there are no repeated symbols in any row or column. Here is an example for when n = 3 : 1 2 3 2 3 1 3 1 2 Given any such square, form its corresponding Latin square graph as follows: create n2 vertices, one for each cell in the square. Connect two vertices by an edge whenever one of the three following conditions hold: • The two cells corresponding to this vertex lie in the same row. • The two cells corresponding to this vertex lie in the same column. • The two cells corresponding to this vertex contain the same symbol. For any n ≥ 3, prove that this graph is regular with degree 3(n − 1). Show that this graph has precisely three distinct eigenvalues in its spectrum. What are they? 1.